In order to calculate the desired matrix elements, we require the cubic and quartic self-interactions of the 5D tensor field \(\hat{h}_{\mu \nu }\)  as well as the \(\hat{h} \hat{h} \hat{r}\)  cubic interaction. The \(\hat{h}\)  self-interactions (and their 4D effective equivalents) are changed from our previous works [1]}, [2]}, [3]} only in the specific choice of \(A(y)\) . Following integration-by-parts and the elimination of total derivatives, we find that the \(\hat{h}\hat{h}\hat{r}\)  interaction is similarly identical to the unstabilized case [4]}:
\(\mathcal {L}_{hhr} &= -\dfrac{\kappa }{2r_{c}^{2}}\sqrt{\dfrac{3}{2}} e^{-2A(\varphi )} \Big [(\hat{h}^{\,\prime })^{2} - \hat{h}^{\,\prime }_{\mu \nu }(\hat{h}^{\mu \nu })^{\,\prime } \Big ] \hat{r}~. \) 
